low end of the # scale than it does at the high end of the wv 
scale. From the properties of the Jacobian, it then follows that 
equation (10.54) holds and Be i is given by both an integration 
x 
over the @ ,@ plane and the y,,0, plane (see Courant [1937]). 
The integration over the 8% coordinate then can be defined to 
yield the function [A( v 598,71 as given by equation (10.55). 
The angle, 0*, is, of course, fixed for each single flight. 
It is now possible to describe the functions which can be 
recorded by current instrumental techniques. First, the free 
surface as a function of time at a fixed point in space can be 
recorded. The function which results is a Gaussian Lebesgue 
Power integral of the form of equation (10.56) @r (7.1)) as has 
been proved at the start of this chapter for the case of a short 
crested sea surface. Secondly, for a fixed 6* the sea surface 
as a function of x' can be recorded. In equation (10.53), if y' 
and t' are fixed, then by exactly the same techniques that were 
employed to study the short crested sea surface as a function of 
time it is possible to prove that the free surface as a function 
of x' for a fixed y' and t is given by equation (10.57). 
Both functions are samples of stationary series and both can 
be analyzed by the methods presented above in order to determine 
[a(n )]? and [A( v9") 1° for a finite net. In addition many dif- 
ferent values of 6* can be chosen and a whole set of functions of 
the form of [A( v.90") 1° for different ©* can be found. Thus the 
observed power spectra are given by equation (10.58). From these 
data, the problem is to find an estimate of [ete (Gu peDiloab 
Boe 
